LMFDB - Embedded newform 1152.3.m.c.415.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1152,3,Mod(415,1152)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1152, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 1, 0]))

N = Newforms(chi, 3, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1152.415");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1152 = 2^{7} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1152.m (of order $4$, degree $2$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$31.3897264543$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(i)$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 6 x^{14} - 4 x^{13} + 10 x^{12} + 56 x^{11} + 88 x^{10} - 128 x^{9} - 496 x^{8} - 512 x^{7} + \cdots + 65536 $ x^16 - 6x^14 - 4x^13 + 10x^12 + 56x^11 + 88x^10 - 128x^9 - 496x^8 - 512x^7 + 1408x^6 + 3584x^5 + 2560x^4 - 4096x^3 - 24576*x^2 + 65536
Coefficient ring:	$\Z[a_1, \ldots, a_{25}]$
Coefficient ring index:	$ 2^{28} $
Twist minimal:	no (minimal twist has level 48)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			415.8
Root			$1.80398 + 0.863518i$ of defining polynomial
Character	$\chi$	$=$	1152.415
Dual form			1152.3.m.c.991.8

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(6.49473 + 6.49473i) q^{5} -3.94273 q^{7} +O(q^{10})$	$q+(6.49473 + 6.49473i) q^{5} -3.94273 q^{7} +(-4.31091 + 4.31091i) q^{11} +(-4.06281 + 4.06281i) q^{13} +14.5538 q^{17} +(4.94805 + 4.94805i) q^{19} -43.6717 q^{23} +59.3629i q^{25} +(25.0979 - 25.0979i) q^{29} +32.5024i q^{31} +(-25.6069 - 25.6069i) q^{35} +(-4.14345 - 4.14345i) q^{37} +55.3348i q^{41} +(-16.1189 + 16.1189i) q^{43} +7.92420i q^{47} -33.4549 q^{49} +(-31.5748 - 31.5748i) q^{53} -55.9964 q^{55} +(49.7172 - 49.7172i) q^{59} +(-44.4711 + 44.4711i) q^{61} -52.7736 q^{65} +(-1.64068 - 1.64068i) q^{67} +24.1145 q^{71} -10.7741i q^{73} +(16.9967 - 16.9967i) q^{77} -72.0517i q^{79} +(-42.0499 - 42.0499i) q^{83} +(94.5229 + 94.5229i) q^{85} -28.9853i q^{89} +(16.0185 - 16.0185i) q^{91} +64.2724i q^{95} -54.2698 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q+O(q^{10}) $ 16 * q	$ 16 q - 32 q^{11} - 32 q^{19} - 128 q^{23} + 32 q^{29} - 96 q^{35} + 96 q^{37} + 160 q^{43} + 112 q^{49} - 160 q^{53} + 256 q^{55} + 128 q^{59} + 32 q^{61} + 32 q^{65} + 320 q^{67} + 512 q^{71} + 224 q^{77} + 160 q^{83} - 160 q^{85} - 480 q^{91}+O(q^{100}) $ 16 * q - 32 * q^11 - 32 * q^19 - 128 * q^23 + 32 * q^29 - 96 * q^35 + 96 * q^37 + 160 * q^43 + 112 * q^49 - 160 * q^53 + 256 * q^55 + 128 * q^59 + 32 * q^61 + 32 * q^65 + 320 * q^67 + 512 * q^71 + 224 * q^77 + 160 * q^83 - 160 * q^85 - 480 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1152\mathbb{Z}\right)^\times$.

$n$	$127$	$641$	$901$
$\chi(n)$	$-1$	$1$	$e\left(\frac{1}{4}\right)$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	6.49473	+	6.49473i	1.29895	+	1.29895i	0.929089	+	0.369856i	$0.120593\pi$
$5$	6.49473	+	6.49473i	1.29895	+	1.29895i	0.369856	+	0.929089i	$0.379407\pi$
$6$		0			0
$7$	−3.94273			−0.563247			−0.281623	−	0.959525i	$-0.590873\pi$
$7$	−3.94273			−0.563247			−0.281623	+	0.959525i	$0.590873\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−4.31091	+	4.31091i	−0.391901	+	0.391901i	−0.875364	−	0.483464i	$-0.839379\pi$
$11$	−4.31091	+	4.31091i	−0.391901	+	0.391901i	0.483464	+	0.875364i	$0.339379\pi$
$12$		0			0
$13$	−4.06281	+	4.06281i	−0.312524	+	0.312524i	−0.845887	−	0.533363i	$-0.820928\pi$
$13$	−4.06281	+	4.06281i	−0.312524	+	0.312524i	0.533363	+	0.845887i	$0.320928\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	14.5538			0.856106			0.428053	−	0.903754i	$-0.359200\pi$
$17$	14.5538			0.856106			0.428053	+	0.903754i	$0.359200\pi$
$18$		0			0
$19$	4.94805	+	4.94805i	0.260423	+	0.260423i	0.825226	−	0.564803i	$-0.191048\pi$
$19$	4.94805	+	4.94805i	0.260423	+	0.260423i	−0.564803	+	0.825226i	$0.691048\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−43.6717			−1.89877			−0.949385	−	0.314115i	$-0.898292\pi$
$23$	−43.6717			−1.89877			−0.949385	+	0.314115i	$0.898292\pi$
$24$		0			0
$25$			59.3629i			2.37452i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	25.0979	−	25.0979i	0.865445	−	0.865445i	−0.126519	−	0.991964i	$-0.540381\pi$
$29$	25.0979	−	25.0979i	0.865445	−	0.865445i	0.991964	+	0.126519i	$0.0403806\pi$
$30$		0			0
$31$			32.5024i			1.04846i	0.851576	+	0.524232i	$0.175648\pi$
$31$			32.5024i			1.04846i	−0.851576	+	0.524232i	$0.824352\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−25.6069	−	25.6069i	−0.731627	−	0.731627i
$36$		0			0
$37$	−4.14345	−	4.14345i	−0.111985	−	0.111985i	0.648894	−	0.760879i	$-0.275232\pi$
$37$	−4.14345	−	4.14345i	−0.111985	−	0.111985i	−0.760879	+	0.648894i	$0.775232\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			55.3348i			1.34963i	0.737987	+	0.674814i	$0.235776\pi$
$41$			55.3348i			1.34963i	−0.737987	+	0.674814i	$0.764224\pi$
$42$		0			0
$43$	−16.1189	+	16.1189i	−0.374858	+	0.374858i	−0.869243	−	0.494385i	$-0.835393\pi$
$43$	−16.1189	+	16.1189i	−0.374858	+	0.374858i	0.494385	+	0.869243i	$0.335393\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			7.92420i			0.168600i	0.996440	+	0.0843001i	$0.0268654\pi$
$47$			7.92420i			0.168600i	−0.996440	+	0.0843001i	$0.973135\pi$
$48$		0			0
$49$	−33.4549			−0.682753
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−31.5748	−	31.5748i	−0.595750	−	0.595750i	0.343429	−	0.939179i	$-0.388412\pi$
$53$	−31.5748	−	31.5748i	−0.595750	−	0.595750i	−0.939179	+	0.343429i	$0.888412\pi$
$54$		0			0
$55$	−55.9964			−1.01812
$56$		0			0
$57$		0			0
$58$		0			0
$59$	49.7172	−	49.7172i	0.842665	−	0.842665i	−0.146540	−	0.989205i	$-0.546814\pi$
$59$	49.7172	−	49.7172i	0.842665	−	0.842665i	0.989205	+	0.146540i	$0.0468137\pi$
$60$		0			0
$61$	−44.4711	+	44.4711i	−0.729035	+	0.729035i	−0.970427	−	0.241393i	$-0.922396\pi$
$61$	−44.4711	+	44.4711i	−0.729035	+	0.729035i	0.241393	+	0.970427i	$0.422396\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−52.7736			−0.811902
$66$		0			0
$67$	−1.64068	−	1.64068i	−0.0244878	−	0.0244878i	0.694757	−	0.719245i	$-0.255512\pi$
$67$	−1.64068	−	1.64068i	−0.0244878	−	0.0244878i	−0.719245	+	0.694757i	$0.755512\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	24.1145			0.339641			0.169821	−	0.985475i	$-0.445681\pi$
$71$	24.1145			0.339641			0.169821	+	0.985475i	$0.445681\pi$
$72$		0			0
$73$		−	10.7741i		−	0.147591i	−0.997273	−	0.0737955i	$-0.976489\pi$
$73$		−	10.7741i		−	0.147591i	0.997273	−	0.0737955i	$-0.0235112\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	16.9967	−	16.9967i	0.220737	−	0.220737i
$78$		0			0
$79$		−	72.0517i		−	0.912047i	−0.889968	−	0.456024i	$-0.849273\pi$
$79$		−	72.0517i		−	0.912047i	0.889968	−	0.456024i	$-0.150727\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−42.0499	−	42.0499i	−0.506625	−	0.506625i	0.406864	−	0.913489i	$-0.366622\pi$
$83$	−42.0499	−	42.0499i	−0.506625	−	0.506625i	−0.913489	+	0.406864i	$0.866622\pi$
$84$		0			0
$85$	94.5229	+	94.5229i	1.11203	+	1.11203i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	28.9853i		−	0.325677i	−0.986653	−	0.162839i	$-0.947935\pi$
$89$		−	28.9853i		−	0.325677i	0.986653	−	0.162839i	$-0.0520650\pi$
$90$		0			0
$91$	16.0185	−	16.0185i	0.176028	−	0.176028i
$92$		0			0
$93$		0			0
$94$		0			0
$95$			64.2724i			0.676552i
$96$		0			0
$97$	−54.2698			−0.559483			−0.279741	−	0.960075i	$-0.590249\pi$
$97$	−54.2698			−0.559483			−0.279741	+	0.960075i	$0.590249\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	57.0829	+	57.0829i	0.565177	+	0.565177i	0.930773	−	0.365597i	$-0.119135\pi$
$101$	57.0829	+	57.0829i	0.565177	+	0.565177i	−0.365597	+	0.930773i	$0.619135\pi$
$102$		0			0
$103$	−39.3048			−0.381600			−0.190800	−	0.981629i	$-0.561108\pi$
$103$	−39.3048			−0.381600			−0.190800	+	0.981629i	$0.561108\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−25.6981	+	25.6981i	−0.240169	+	0.240169i	−0.816920	−	0.576751i	$-0.804320\pi$
$107$	−25.6981	+	25.6981i	−0.240169	+	0.240169i	0.576751	+	0.816920i	$0.304320\pi$
$108$		0			0
$109$	9.66133	−	9.66133i	0.0886360	−	0.0886360i	−0.661399	−	0.750035i	$-0.730037\pi$
$109$	9.66133	−	9.66133i	0.0886360	−	0.0886360i	0.750035	+	0.661399i	$0.230037\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−64.2927			−0.568962			−0.284481	−	0.958682i	$-0.591821\pi$
$113$	−64.2927			−0.568962			−0.284481	+	0.958682i	$0.591821\pi$
$114$		0			0
$115$	−283.636	−	283.636i	−2.46640	−	2.46640i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−57.3816			−0.482199
$120$		0			0
$121$			83.8321i			0.692827i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−223.178	+	223.178i	−1.78542	+	1.78542i
$126$		0			0
$127$			129.668i			1.02101i	0.859875	+	0.510504i	$0.170541\pi$
$127$			129.668i			1.02101i	−0.859875	+	0.510504i	$0.829459\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	118.504	+	118.504i	0.904613	+	0.904613i	0.995831	−	0.0912183i	$-0.0290761\pi$
$131$	118.504	+	118.504i	0.904613	+	0.904613i	−0.0912183	+	0.995831i	$0.529076\pi$
$132$		0			0
$133$	−19.5088	−	19.5088i	−0.146683	−	0.146683i
$134$		0			0
$135$		0			0
$136$		0			0
$137$			157.472i			1.14943i	0.818353	+	0.574716i	$0.194888\pi$
$137$			157.472i			1.14943i	−0.818353	+	0.574716i	$0.805112\pi$
$138$		0			0
$139$	−118.943	+	118.943i	−0.855703	+	0.855703i	−0.990829	−	0.135125i	$-0.956856\pi$
$139$	−118.943	+	118.943i	−0.855703	+	0.855703i	0.135125	+	0.990829i	$0.456856\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		−	35.0288i		−	0.244957i
$144$		0			0
$145$	326.008			2.24833
$146$		0			0
$147$		0			0
$148$		0			0
$149$	99.5402	+	99.5402i	0.668055	+	0.668055i	0.957266	−	0.289210i	$-0.0933927\pi$
$149$	99.5402	+	99.5402i	0.668055	+	0.668055i	−0.289210	+	0.957266i	$0.593393\pi$
$150$		0			0
$151$	−273.705			−1.81262			−0.906308	−	0.422618i	$-0.861111\pi$
$151$	−273.705			−1.81262			−0.906308	+	0.422618i	$0.861111\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−211.094	+	211.094i	−1.36190	+	1.36190i
$156$		0			0
$157$	75.8792	−	75.8792i	0.483307	−	0.483307i	−0.422879	−	0.906186i	$-0.638981\pi$
$157$	75.8792	−	75.8792i	0.483307	−	0.483307i	0.906186	+	0.422879i	$0.138981\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	172.186			1.06948
$162$		0			0
$163$	177.242	+	177.242i	1.08737	+	1.08737i	0.995798	+	0.0915766i	$0.0291906\pi$
$163$	177.242	+	177.242i	1.08737	+	1.08737i	0.0915766	+	0.995798i	$0.470809\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	61.6774			0.369326			0.184663	−	0.982802i	$-0.440881\pi$
$167$	61.6774			0.369326			0.184663	+	0.982802i	$0.440881\pi$
$168$		0			0
$169$			135.987i			0.804658i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	69.7012	−	69.7012i	0.402897	−	0.402897i	−0.476355	−	0.879253i	$-0.658042\pi$
$173$	69.7012	−	69.7012i	0.402897	−	0.402897i	0.879253	+	0.476355i	$0.158042\pi$
$174$		0			0
$175$		−	234.052i		−	1.33744i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−43.6228	−	43.6228i	−0.243703	−	0.243703i	0.574677	−	0.818380i	$-0.305128\pi$
$179$	−43.6228	−	43.6228i	−0.243703	−	0.243703i	−0.818380	+	0.574677i	$0.805128\pi$
$180$		0			0
$181$	44.7291	+	44.7291i	0.247122	+	0.247122i	0.819788	−	0.572666i	$-0.194091\pi$
$181$	44.7291	+	44.7291i	0.247122	+	0.247122i	−0.572666	+	0.819788i	$0.694091\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		−	53.8211i		−	0.290925i
$186$		0			0
$187$	−62.7401	+	62.7401i	−0.335509	+	0.335509i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	171.759i		−	0.899263i	−0.893214	−	0.449632i	$-0.851555\pi$
$191$		−	171.759i		−	0.899263i	0.893214	−	0.449632i	$-0.148445\pi$
$192$		0			0
$193$	−215.384			−1.11598			−0.557989	−	0.829848i	$-0.688427\pi$
$193$	−215.384			−1.11598			−0.557989	+	0.829848i	$0.688427\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−18.3354	−	18.3354i	−0.0930731	−	0.0930731i	0.659037	−	0.752110i	$-0.270964\pi$
$197$	−18.3354	−	18.3354i	−0.0930731	−	0.0930731i	−0.752110	+	0.659037i	$0.770964\pi$
$198$		0			0
$199$	227.112			1.14127			0.570634	−	0.821205i	$-0.306698\pi$
$199$	227.112			1.14127			0.570634	+	0.821205i	$0.306698\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−98.9542	+	98.9542i	−0.487459	+	0.487459i
$204$		0			0
$205$	−359.384	+	359.384i	−1.75309	+	1.75309i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−42.6612			−0.204120
$210$		0			0
$211$	190.206	+	190.206i	0.901451	+	0.901451i	0.995562	−	0.0941112i	$-0.0300009\pi$
$211$	190.206	+	190.206i	0.901451	+	0.901451i	−0.0941112	+	0.995562i	$0.530001\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−209.375			−0.973839
$216$		0			0
$217$		−	128.148i		−	0.590544i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−59.1293	+	59.1293i	−0.267553	+	0.267553i
$222$		0			0
$223$		−	154.401i		−	0.692379i	−0.938165	−	0.346190i	$-0.887475\pi$
$223$		−	154.401i		−	0.692379i	0.938165	−	0.346190i	$-0.112525\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	36.8204	+	36.8204i	0.162204	+	0.162204i	0.783543	−	0.621338i	$-0.213411\pi$
$227$	36.8204	+	36.8204i	0.162204	+	0.162204i	−0.621338	+	0.783543i	$0.713411\pi$
$228$		0			0
$229$	−17.9692	−	17.9692i	−0.0784683	−	0.0784683i	0.666783	−	0.745252i	$-0.267671\pi$
$229$	−17.9692	−	17.9692i	−0.0784683	−	0.0784683i	−0.745252	+	0.666783i	$0.767671\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	167.669i		−	0.719608i	−0.933028	−	0.359804i	$-0.882844\pi$
$233$		−	167.669i		−	0.719608i	0.933028	−	0.359804i	$-0.117156\pi$
$234$		0			0
$235$	−51.4655	+	51.4655i	−0.219002	+	0.219002i
$236$		0			0
$237$		0			0
$238$		0			0
$239$			29.7509i			0.124481i	0.998061	+	0.0622403i	$0.0198245\pi$
$239$			29.7509i			0.124481i	−0.998061	+	0.0622403i	$0.980175\pi$
$240$		0			0
$241$	−107.373			−0.445531			−0.222766	−	0.974872i	$-0.571508\pi$
$241$	−107.373			−0.445531			−0.222766	+	0.974872i	$0.571508\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−217.280	−	217.280i	−0.886859	−	0.886859i
$246$		0			0
$247$	−40.2059			−0.162777
$248$		0			0
$249$		0			0
$250$		0			0
$251$	342.946	−	342.946i	1.36632	−	1.36632i	0.500697	−	0.865623i	$-0.333077\pi$
$251$	342.946	−	342.946i	1.36632	−	1.36632i	0.865623	−	0.500697i	$-0.166923\pi$
$252$		0			0
$253$	188.265	−	188.265i	0.744130	−	0.744130i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	393.565			1.53138			0.765691	−	0.643209i	$-0.222397\pi$
$257$	393.565			1.53138			0.765691	+	0.643209i	$0.222397\pi$
$258$		0			0
$259$	16.3365	+	16.3365i	0.0630753	+	0.0630753i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	413.800			1.57338			0.786692	−	0.617346i	$-0.211792\pi$
$263$	413.800			1.57338			0.786692	+	0.617346i	$0.211792\pi$
$264$		0			0
$265$		−	410.139i		−	1.54769i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	165.389	−	165.389i	0.614830	−	0.614830i	−0.329371	−	0.944201i	$-0.606837\pi$
$269$	165.389	−	165.389i	0.614830	−	0.614830i	0.944201	+	0.329371i	$0.106837\pi$
$270$		0			0
$271$			309.821i			1.14325i	0.820514	+	0.571626i	$0.193687\pi$
$271$			309.821i			1.14325i	−0.820514	+	0.571626i	$0.806313\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−255.908	−	255.908i	−0.930575	−	0.930575i
$276$		0			0
$277$	−157.397	−	157.397i	−0.568221	−	0.568221i	0.363409	−	0.931630i	$-0.381613\pi$
$277$	−157.397	−	157.397i	−0.568221	−	0.568221i	−0.931630	+	0.363409i	$0.881613\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	411.141i		−	1.46313i	−0.681769	−	0.731567i	$-0.738789\pi$
$281$		−	411.141i		−	1.46313i	0.681769	−	0.731567i	$-0.261211\pi$
$282$		0			0
$283$	343.521	−	343.521i	1.21385	−	1.21385i	0.244106	−	0.969748i	$-0.421505\pi$
$283$	343.521	−	343.521i	1.21385	−	1.21385i	0.969748	−	0.244106i	$-0.0784946\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		−	218.170i		−	0.760174i
$288$		0			0
$289$	−77.1870			−0.267083
$290$		0			0
$291$		0			0
$292$		0			0
$293$	35.1386	+	35.1386i	0.119927	+	0.119927i	0.764523	−	0.644596i	$-0.222975\pi$
$293$	35.1386	+	35.1386i	0.119927	+	0.119927i	−0.644596	+	0.764523i	$0.722975\pi$
$294$		0			0
$295$	645.799			2.18915
$296$		0			0
$297$		0			0
$298$		0			0
$299$	177.430	−	177.430i	0.593410	−	0.593410i
$300$		0			0
$301$	63.5524	−	63.5524i	0.211137	−	0.211137i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−577.655			−1.89395
$306$		0			0
$307$	−16.4432	−	16.4432i	−0.0535609	−	0.0535609i	0.679819	−	0.733380i	$-0.262058\pi$
$307$	−16.4432	−	16.4432i	−0.0535609	−	0.0535609i	−0.733380	+	0.679819i	$0.762058\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	39.8016			0.127980			0.0639898	−	0.997951i	$-0.479618\pi$
$311$	39.8016			0.127980			0.0639898	+	0.997951i	$0.479618\pi$
$312$		0			0
$313$			431.885i			1.37982i	0.723894	+	0.689911i	$0.242351\pi$
$313$			431.885i			1.37982i	−0.723894	+	0.689911i	$0.757649\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	255.063	−	255.063i	0.804615	−	0.804615i	−0.179198	−	0.983813i	$-0.557350\pi$
$317$	255.063	−	255.063i	0.804615	−	0.804615i	0.983813	+	0.179198i	$0.0573503\pi$
$318$		0			0
$319$			216.390i			0.678337i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	72.0128	+	72.0128i	0.222950	+	0.222950i
$324$		0			0
$325$	−241.180	−	241.180i	−0.742092	−	0.742092i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		−	31.2430i		−	0.0949635i
$330$		0			0
$331$	205.897	−	205.897i	0.622045	−	0.622045i	−0.324009	−	0.946054i	$-0.605031\pi$
$331$	205.897	−	205.897i	0.622045	−	0.622045i	0.946054	+	0.324009i	$0.105031\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		−	21.3115i		−	0.0636165i
$336$		0			0
$337$	45.7312			0.135701			0.0678504	−	0.997696i	$-0.478386\pi$
$337$	45.7312			0.135701			0.0678504	+	0.997696i	$0.478386\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−140.115	−	140.115i	−0.410894	−	0.410894i
$342$		0			0
$343$	325.097			0.947805
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−296.512	+	296.512i	−0.854500	+	0.854500i	−0.990684	−	0.136183i	$-0.956516\pi$
$347$	−296.512	+	296.512i	−0.854500	+	0.854500i	0.136183	+	0.990684i	$0.456516\pi$
$348$		0			0
$349$	198.107	−	198.107i	0.567641	−	0.567641i	−0.363826	−	0.931467i	$-0.618530\pi$
$349$	198.107	−	198.107i	0.567641	−	0.567641i	0.931467	+	0.363826i	$0.118530\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−85.4490			−0.242065			−0.121033	−	0.992649i	$-0.538621\pi$
$353$	−85.4490			−0.242065			−0.121033	+	0.992649i	$0.538621\pi$
$354$		0			0
$355$	156.617	+	156.617i	0.441176	+	0.441176i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−302.214			−0.841823			−0.420911	−	0.907102i	$-0.638290\pi$
$359$	−302.214			−0.841823			−0.420911	+	0.907102i	$0.638290\pi$
$360$		0			0
$361$		−	312.034i		−	0.864359i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	69.9751	−	69.9751i	0.191713	−	0.191713i
$366$		0			0
$367$			372.554i			1.01513i	0.861612	+	0.507567i	$0.169455\pi$
$367$			372.554i			1.01513i	−0.861612	+	0.507567i	$0.830545\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	124.491	+	124.491i	0.335554	+	0.335554i
$372$		0			0
$373$	407.130	+	407.130i	1.09150	+	1.09150i	0.995369	+	0.0961318i	$0.0306470\pi$
$373$	407.130	+	407.130i	1.09150	+	1.09150i	0.0961318	+	0.995369i	$0.469353\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			203.936i			0.540944i
$378$		0			0
$379$	−117.854	+	117.854i	−0.310961	+	0.310961i	−0.845282	−	0.534321i	$-0.820567\pi$
$379$	−117.854	+	117.854i	−0.310961	+	0.310961i	0.534321	+	0.845282i	$0.320567\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$			407.983i			1.06523i	0.846357	+	0.532615i	$0.178791\pi$
$383$			407.983i			1.06523i	−0.846357	+	0.532615i	$0.821209\pi$
$384$		0			0
$385$	220.778			0.573450
$386$		0			0
$387$		0			0
$388$		0			0
$389$	458.508	+	458.508i	1.17868	+	1.17868i	0.980080	+	0.198605i	$0.0636411\pi$
$389$	458.508	+	458.508i	1.17868	+	1.17868i	0.198605	+	0.980080i	$0.436359\pi$
$390$		0			0
$391$	−635.589			−1.62555
$392$		0			0
$393$		0			0
$394$		0			0
$395$	467.956	−	467.956i	1.18470	−	1.18470i
$396$		0			0
$397$	259.865	−	259.865i	0.654573	−	0.654573i	−0.299518	−	0.954091i	$-0.596826\pi$
$397$	259.865	−	259.865i	0.654573	−	0.654573i	0.954091	+	0.299518i	$0.0968259\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	499.197			1.24488			0.622441	−	0.782667i	$-0.286141\pi$
$401$	499.197			1.24488			0.622441	+	0.782667i	$0.286141\pi$
$402$		0			0
$403$	−132.051	−	132.051i	−0.327670	−	0.327670i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	35.7241			0.0877741
$408$		0			0
$409$		−	494.949i		−	1.21014i	−0.796171	−	0.605072i	$-0.793144\pi$
$409$		−	494.949i		−	1.21014i	0.796171	−	0.605072i	$-0.206856\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−196.021	+	196.021i	−0.474628	+	0.474628i
$414$		0			0
$415$		−	546.205i		−	1.31616i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	560.555	+	560.555i	1.33784	+	1.33784i	0.898148	+	0.439693i	$0.144913\pi$
$419$	560.555	+	560.555i	1.33784	+	1.33784i	0.439693	+	0.898148i	$0.355087\pi$
$420$		0			0
$421$	−397.946	−	397.946i	−0.945239	−	0.945239i	0.0533373	−	0.998577i	$-0.483014\pi$
$421$	−397.946	−	397.946i	−0.945239	−	0.945239i	−0.998577	+	0.0533373i	$0.983014\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$			863.956i			2.03284i
$426$		0			0
$427$	175.337	−	175.337i	0.410626	−	0.410626i
$428$		0			0
$429$		0			0
$430$		0			0
$431$			662.874i			1.53799i	0.639255	+	0.768995i	$0.279243\pi$
$431$			662.874i			1.53799i	−0.639255	+	0.768995i	$0.720757\pi$
$432$		0			0
$433$	−338.800			−0.782448			−0.391224	−	0.920296i	$-0.627948\pi$
$433$	−338.800			−0.782448			−0.391224	+	0.920296i	$0.627948\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−216.090	−	216.090i	−0.494484	−	0.494484i
$438$		0			0
$439$	234.566			0.534319			0.267160	−	0.963652i	$-0.413915\pi$
$439$	234.566			0.534319			0.267160	+	0.963652i	$0.413915\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−421.096	+	421.096i	−0.950555	+	0.950555i	−0.998834	−	0.0482792i	$-0.984626\pi$
$443$	−421.096	+	421.096i	−0.950555	+	0.950555i	0.0482792	+	0.998834i	$0.484626\pi$
$444$		0			0
$445$	188.251	−	188.251i	0.423037	−	0.423037i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−492.636			−1.09718			−0.548592	−	0.836090i	$-0.684836\pi$
$449$	−492.636			−1.09718			−0.548592	+	0.836090i	$0.684836\pi$
$450$		0			0
$451$	−238.543	−	238.543i	−0.528921	−	0.528921i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	208.072			0.457301
$456$		0			0
$457$		−	516.831i		−	1.13092i	−0.824775	−	0.565461i	$-0.808698\pi$
$457$		−	516.831i		−	1.13092i	0.824775	−	0.565461i	$-0.191302\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−27.5260	+	27.5260i	−0.0597093	+	0.0597093i	−0.736331	−	0.676622i	$-0.763443\pi$
$461$	−27.5260	+	27.5260i	−0.0597093	+	0.0597093i	0.676622	+	0.736331i	$0.263443\pi$
$462$		0			0
$463$		−	122.111i		−	0.263740i	−0.991267	−	0.131870i	$-0.957902\pi$
$463$		−	122.111i		−	0.263740i	0.991267	−	0.131870i	$-0.0420981\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−267.964	−	267.964i	−0.573798	−	0.573798i	0.359390	−	0.933188i	$-0.382985\pi$
$467$	−267.964	−	267.964i	−0.573798	−	0.573798i	−0.933188	+	0.359390i	$0.882985\pi$
$468$		0			0
$469$	6.46875	+	6.46875i	0.0137926	+	0.0137926i
$470$		0			0
$471$		0			0
$472$		0			0
$473$		−	138.974i		−	0.293814i
$474$		0			0
$475$	−293.730	+	293.730i	−0.618380	+	0.618380i
$476$		0			0
$477$		0			0
$478$		0			0
$479$			419.084i			0.874915i	0.899239	+	0.437457i	$0.144121\pi$
$479$			419.084i			0.874915i	−0.899239	+	0.437457i	$0.855879\pi$
$480$		0			0
$481$	33.6681			0.0699960
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−352.468	−	352.468i	−0.726738	−	0.726738i
$486$		0			0
$487$	57.2378			0.117531			0.0587657	−	0.998272i	$-0.481284\pi$
$487$	57.2378			0.117531			0.0587657	+	0.998272i	$0.481284\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	301.955	−	301.955i	0.614979	−	0.614979i	−0.329260	−	0.944239i	$-0.606799\pi$
$491$	301.955	−	301.955i	0.614979	−	0.614979i	0.944239	+	0.329260i	$0.106799\pi$
$492$		0			0
$493$	365.270	−	365.270i	0.740912	−	0.740912i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−95.0771			−0.191302
$498$		0			0
$499$	619.990	+	619.990i	1.24247	+	1.24247i	0.958975	+	0.283491i	$0.0914925\pi$
$499$	619.990	+	619.990i	1.24247	+	1.24247i	0.283491	+	0.958975i	$0.408507\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	222.446			0.442239			0.221120	−	0.975247i	$-0.429029\pi$
$503$	222.446			0.442239			0.221120	+	0.975247i	$0.429029\pi$
$504$		0			0
$505$			741.475i			1.46827i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	489.873	−	489.873i	0.962421	−	0.962421i	−0.0368976	−	0.999319i	$-0.511748\pi$
$509$	489.873	−	489.873i	0.962421	−	0.962421i	0.999319	+	0.0368976i	$0.0117475\pi$
$510$		0			0
$511$			42.4795i			0.0831302i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−255.274	−	255.274i	−0.495678	−	0.495678i
$516$		0			0
$517$	−34.1605	−	34.1605i	−0.0660745	−	0.0660745i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	197.152i		−	0.378412i	−0.981937	−	0.189206i	$-0.939409\pi$
$521$		−	197.152i		−	0.378412i	0.981937	−	0.189206i	$-0.0605913\pi$
$522$		0			0
$523$	621.874	−	621.874i	1.18905	−	1.18905i	0.211721	−	0.977330i	$-0.432093\pi$
$523$	621.874	−	621.874i	1.18905	−	1.18905i	0.977330	−	0.211721i	$-0.0679067\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			473.033i			0.897596i
$528$		0			0
$529$	1378.22			2.60533
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−224.815	−	224.815i	−0.421791	−	0.421791i
$534$		0			0
$535$	−333.804			−0.623933
$536$		0			0
$537$		0			0
$538$		0			0
$539$	144.221	−	144.221i	0.267572	−	0.267572i
$540$		0			0
$541$	−423.563	+	423.563i	−0.782925	+	0.782925i	−0.980323	−	0.197398i	$-0.936751\pi$
$541$	−423.563	+	423.563i	−0.782925	+	0.782925i	0.197398	+	0.980323i	$0.436751\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	125.495			0.230267
$546$		0			0
$547$	−14.5553	−	14.5553i	−0.0266093	−	0.0266093i	0.693677	−	0.720286i	$-0.255990\pi$
$547$	−14.5553	−	14.5553i	−0.0266093	−	0.0266093i	−0.720286	+	0.693677i	$0.755990\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	248.371			0.450764
$552$		0			0
$553$			284.080i			0.513708i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−351.991	+	351.991i	−0.631941	+	0.631941i	−0.948554	−	0.316614i	$-0.897454\pi$
$557$	−351.991	+	351.991i	−0.631941	+	0.631941i	0.316614	+	0.948554i	$0.397454\pi$
$558$		0			0
$559$		−	130.976i		−	0.234304i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	150.902	+	150.902i	0.268031	+	0.268031i	0.828307	−	0.560275i	$-0.189305\pi$
$563$	150.902	+	150.902i	0.268031	+	0.268031i	−0.560275	+	0.828307i	$0.689305\pi$
$564$		0			0
$565$	−417.563	−	417.563i	−0.739050	−	0.739050i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	113.300i		−	0.199121i	−0.995032	−	0.0995603i	$-0.968256\pi$
$569$		−	113.300i		−	0.199121i	0.995032	−	0.0995603i	$-0.0317436\pi$
$570$		0			0
$571$	−207.486	+	207.486i	−0.363373	+	0.363373i	−0.865053	−	0.501680i	$-0.832715\pi$
$571$	−207.486	+	207.486i	−0.363373	+	0.363373i	0.501680	+	0.865053i	$0.332715\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		−	2592.48i		−	4.50866i
$576$		0			0
$577$	−484.715			−0.840061			−0.420031	−	0.907510i	$-0.637981\pi$
$577$	−484.715			−0.840061			−0.420031	+	0.907510i	$0.637981\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	165.791	+	165.791i	0.285355	+	0.285355i
$582$		0			0
$583$	272.232			0.466950
$584$		0			0
$585$		0			0
$586$		0			0
$587$	540.404	−	540.404i	0.920619	−	0.920619i	−0.0764537	−	0.997073i	$-0.524360\pi$
$587$	540.404	−	540.404i	0.920619	−	0.920619i	0.997073	+	0.0764537i	$0.0243597\pi$
$588$		0			0
$589$	−160.823	+	160.823i	−0.273045	+	0.273045i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−411.176			−0.693383			−0.346692	−	0.937979i	$-0.612695\pi$
$593$	−411.176			−0.693383			−0.346692	+	0.937979i	$0.612695\pi$
$594$		0			0
$595$	−372.678	−	372.678i	−0.626350	−	0.626350i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−552.839			−0.922936			−0.461468	−	0.887157i	$-0.652677\pi$
$599$	−552.839			−0.922936			−0.461468	+	0.887157i	$0.652677\pi$
$600$		0			0
$601$		−	881.159i		−	1.46615i	−0.680145	−	0.733077i	$-0.738083\pi$
$601$		−	881.159i		−	1.46615i	0.680145	−	0.733077i	$-0.261917\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−544.467	+	544.467i	−0.899945	+	0.899945i
$606$		0			0
$607$		−	1175.08i		−	1.93588i	−0.251186	−	0.967939i	$-0.580820\pi$
$607$		−	1175.08i		−	1.93588i	0.251186	−	0.967939i	$-0.419180\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−32.1945	−	32.1945i	−0.0526915	−	0.0526915i
$612$		0			0
$613$	496.928	+	496.928i	0.810649	+	0.810649i	0.984731	−	0.174082i	$-0.0556959\pi$
$613$	496.928	+	496.928i	0.810649	+	0.810649i	−0.174082	+	0.984731i	$0.555696\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	623.301i		−	1.01021i	−0.863057	−	0.505106i	$-0.831453\pi$
$617$		−	623.301i		−	1.01021i	0.863057	−	0.505106i	$-0.168547\pi$
$618$		0			0
$619$	7.45302	−	7.45302i	0.0120404	−	0.0120404i	−0.701061	−	0.713101i	$-0.747290\pi$
$619$	7.45302	−	7.45302i	0.0120404	−	0.0120404i	0.713101	+	0.701061i	$0.247290\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$			114.281i			0.183437i
$624$		0			0
$625$	−1414.88			−2.26381
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−60.3029	−	60.3029i	−0.0958711	−	0.0958711i
$630$		0			0
$631$	−147.833			−0.234284			−0.117142	−	0.993115i	$-0.537373\pi$
$631$	−147.833			−0.234284			−0.117142	+	0.993115i	$0.537373\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−842.158	+	842.158i	−1.32623	+	1.32623i
$636$		0			0
$637$	135.921	−	135.921i	0.213376	−	0.213376i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	782.691			1.22105			0.610523	−	0.791998i	$-0.290959\pi$
$641$	782.691			1.22105			0.610523	+	0.791998i	$0.290959\pi$
$642$		0			0
$643$	126.760	+	126.760i	0.197138	+	0.197138i	0.798772	−	0.601634i	$-0.205483\pi$
$643$	126.760	+	126.760i	0.197138	+	0.197138i	−0.601634	+	0.798772i	$0.705483\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	1226.09			1.89504			0.947520	−	0.319697i	$-0.103581\pi$
$647$	1226.09			1.89504			0.947520	+	0.319697i	$0.103581\pi$
$648$		0			0
$649$			428.653i			0.660482i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	326.300	−	326.300i	0.499694	−	0.499694i	−0.411649	−	0.911343i	$-0.635047\pi$
$653$	326.300	−	326.300i	0.499694	−	0.499694i	0.911343	+	0.411649i	$0.135047\pi$
$654$		0			0
$655$			1539.31i			2.35008i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	574.901	+	574.901i	0.872384	+	0.872384i	0.992732	−	0.120347i	$-0.0384009\pi$
$659$	574.901	+	574.901i	0.872384	+	0.872384i	−0.120347	+	0.992732i	$0.538401\pi$
$660$		0			0
$661$	−52.8795	−	52.8795i	−0.0799993	−	0.0799993i	0.665975	−	0.745974i	$-0.268016\pi$
$661$	−52.8795	−	52.8795i	−0.0799993	−	0.0799993i	−0.745974	+	0.665975i	$0.768016\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		−	253.409i		−	0.381065i
$666$		0			0
$667$	−1096.07	+	1096.07i	−1.64328	+	1.64328i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		−	383.422i		−	0.571419i
$672$		0			0
$673$	342.318			0.508645			0.254322	−	0.967119i	$-0.418148\pi$
$673$	342.318			0.508645			0.254322	+	0.967119i	$0.418148\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	107.154	+	107.154i	0.158278	+	0.158278i	0.781803	−	0.623525i	$-0.214300\pi$
$677$	107.154	+	107.154i	0.158278	+	0.158278i	−0.623525	+	0.781803i	$0.714300\pi$
$678$		0			0
$679$	213.971			0.315127
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−724.233	+	724.233i	−1.06037	+	1.06037i	−0.0623142	+	0.998057i	$0.519848\pi$
$683$	−724.233	+	724.233i	−1.06037	+	1.06037i	−0.998057	+	0.0623142i	$0.980152\pi$
$684$		0			0
$685$	−1022.74	+	1022.74i	−1.49305	+	1.49305i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	256.564			0.372372
$690$		0			0
$691$	162.528	+	162.528i	0.235207	+	0.235207i	0.814862	−	0.579655i	$-0.196813\pi$
$691$	162.528	+	162.528i	0.235207	+	0.235207i	−0.579655	+	0.814862i	$0.696813\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−1545.00			−2.22302
$696$		0			0
$697$			805.331i			1.15543i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−301.659	+	301.659i	−0.430327	+	0.430327i	−0.888740	−	0.458412i	$-0.848418\pi$
$701$	−301.659	+	301.659i	−0.430327	+	0.430327i	0.458412	+	0.888740i	$0.348418\pi$
$702$		0			0
$703$		−	41.0040i		−	0.0583271i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−225.062	−	225.062i	−0.318334	−	0.318334i
$708$		0			0
$709$	629.100	+	629.100i	0.887306	+	0.887306i	0.994264	−	0.106958i	$-0.0341109\pi$
$709$	629.100	+	629.100i	0.887306	+	0.887306i	−0.106958	+	0.994264i	$0.534111\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	1419.43i		−	1.99079i
$714$		0			0
$715$	227.502	−	227.502i	0.318185	−	0.318185i
$716$		0			0
$717$		0			0
$718$		0			0
$719$		−	145.542i		−	0.202422i	−0.994865	−	0.101211i	$-0.967728\pi$
$719$		−	145.542i		−	0.202422i	0.994865	−	0.101211i	$-0.0322718\pi$
$720$		0			0
$721$	154.968			0.214935
$722$		0			0
$723$		0			0
$724$		0			0
$725$	1489.88	+	1489.88i	2.05501	+	2.05501i
$726$		0			0
$727$	938.214			1.29053			0.645264	−	0.763960i	$-0.276747\pi$
$727$	938.214			1.29053			0.645264	+	0.763960i	$0.276747\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−234.591	+	234.591i	−0.320918	+	0.320918i
$732$		0			0
$733$	−692.101	+	692.101i	−0.944203	+	0.944203i	−0.998524	−	0.0543203i	$-0.982701\pi$
$733$	−692.101	+	692.101i	−0.944203	+	0.944203i	0.0543203	+	0.998524i	$0.482701\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	14.1456			0.0191935
$738$		0			0
$739$	−440.389	−	440.389i	−0.595926	−	0.595926i	0.343300	−	0.939226i	$-0.388455\pi$
$739$	−440.389	−	440.389i	−0.595926	−	0.595926i	−0.939226	+	0.343300i	$0.888455\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−1010.54			−1.36008			−0.680039	−	0.733176i	$-0.738037\pi$
$743$	−1010.54			−1.36008			−0.680039	+	0.733176i	$0.738037\pi$
$744$		0			0
$745$			1292.97i			1.73553i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	101.321	−	101.321i	0.135275	−	0.135275i
$750$		0			0
$751$			776.971i			1.03458i	0.855810	+	0.517291i	$0.173060\pi$
$751$			776.971i			1.03458i	−0.855810	+	0.517291i	$0.826940\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−1777.64	−	1777.64i	−2.35449	−	2.35449i
$756$		0			0
$757$	−375.481	−	375.481i	−0.496012	−	0.496012i	0.414182	−	0.910194i	$-0.364068\pi$
$757$	−375.481	−	375.481i	−0.496012	−	0.496012i	−0.910194	+	0.414182i	$0.864068\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			1502.22i			1.97400i	0.160711	+	0.987001i	$0.448621\pi$
$761$			1502.22i			1.97400i	−0.160711	+	0.987001i	$0.551379\pi$
$762$		0			0
$763$	−38.0920	+	38.0920i	−0.0499239	+	0.0499239i
$764$		0			0
$765$		0			0
$766$		0			0
$767$			403.983i			0.526705i
$768$		0			0
$769$	−293.930			−0.382223			−0.191112	−	0.981568i	$-0.561209\pi$
$769$	−293.930			−0.382223			−0.191112	+	0.981568i	$0.561209\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−748.271	−	748.271i	−0.968009	−	0.968009i	0.0314945	−	0.999504i	$-0.489973\pi$
$773$	−748.271	−	748.271i	−0.968009	−	0.968009i	−0.999504	+	0.0314945i	$0.989973\pi$
$774$		0			0
$775$	−1929.44			−2.48960
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−273.799	+	273.799i	−0.351475	+	0.351475i
$780$		0			0
$781$	−103.956	+	103.956i	−0.133106	+	0.133106i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	985.629			1.25558
$786$		0			0
$787$	−735.839	−	735.839i	−0.934992	−	0.934992i	0.0630203	−	0.998012i	$-0.479927\pi$
$787$	−735.839	−	735.839i	−0.934992	−	0.934992i	−0.998012	+	0.0630203i	$0.979927\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	253.488			0.320466
$792$		0			0
$793$		−	361.355i		−	0.455681i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	212.043	−	212.043i	0.266051	−	0.266051i	−0.561455	−	0.827507i	$-0.689758\pi$
$797$	212.043	−	212.043i	0.266051	−	0.266051i	0.827507	+	0.561455i	$0.189758\pi$
$798$		0			0
$799$			115.327i			0.144340i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	46.4464	+	46.4464i	0.0578411	+	0.0578411i
$804$		0			0
$805$	1118.30	+	1118.30i	1.38919	+	1.38919i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			827.491i			1.02286i	0.859326	+	0.511428i	$0.170883\pi$
$809$			827.491i			1.02286i	−0.859326	+	0.511428i	$0.829117\pi$
$810$		0			0
$811$	112.908	−	112.908i	0.139221	−	0.139221i	−0.634062	−	0.773283i	$-0.718613\pi$
$811$	112.908	−	112.908i	0.139221	−	0.139221i	0.773283	+	0.634062i	$0.218613\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$			2302.28i			2.82488i
$816$		0			0
$817$	−159.514			−0.195243
$818$		0			0
$819$		0			0
$820$		0			0
$821$	555.298	+	555.298i	0.676368	+	0.676368i	0.959176	−	0.282808i	$-0.0912661\pi$
$821$	555.298	+	555.298i	0.676368	+	0.676368i	−0.282808	+	0.959176i	$0.591266\pi$
$822$		0			0
$823$	−763.799			−0.928067			−0.464034	−	0.885818i	$-0.653598\pi$
$823$	−763.799			−0.928067			−0.464034	+	0.885818i	$0.653598\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−116.947	+	116.947i	−0.141411	+	0.141411i	−0.774268	−	0.632858i	$-0.781882\pi$
$827$	−116.947	+	116.947i	−0.141411	+	0.141411i	0.632858	+	0.774268i	$0.281882\pi$
$828$		0			0
$829$	−181.419	+	181.419i	−0.218841	+	0.218841i	−0.808010	−	0.589169i	$-0.799455\pi$
$829$	−181.419	+	181.419i	−0.218841	+	0.218841i	0.589169	+	0.808010i	$0.299455\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−486.896			−0.584509
$834$		0			0
$835$	400.578	+	400.578i	0.479734	+	0.479734i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	521.472			0.621540			0.310770	−	0.950485i	$-0.399413\pi$
$839$	521.472			0.621540			0.310770	+	0.950485i	$0.399413\pi$
$840$		0			0
$841$		−	418.809i		−	0.497989i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−883.200	+	883.200i	−1.04521	+	1.04521i
$846$		0			0
$847$		−	330.527i		−	0.390233i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	180.952	+	180.952i	0.212634	+	0.212634i
$852$		0			0
$853$	644.278	+	644.278i	0.755309	+	0.755309i	0.975465	−	0.220156i	$-0.0706566\pi$
$853$	644.278	+	644.278i	0.755309	+	0.755309i	−0.220156	+	0.975465i	$0.570657\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			757.190i			0.883536i	0.897129	+	0.441768i	$0.145649\pi$
$857$			757.190i			0.883536i	−0.897129	+	0.441768i	$0.854351\pi$
$858$		0			0
$859$	−1011.32	+	1011.32i	−1.17732	+	1.17732i	−0.196894	+	0.980425i	$0.563085\pi$
$859$	−1011.32	+	1011.32i	−1.17732	+	1.17732i	−0.980425	+	0.196894i	$0.936915\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	598.188i		−	0.693149i	−0.938022	−	0.346575i	$-0.887345\pi$
$863$		−	598.188i		−	0.693149i	0.938022	−	0.346575i	$-0.112655\pi$
$864$		0			0
$865$	905.381			1.04668
$866$		0			0
$867$		0			0
$868$		0			0
$869$	310.609	+	310.609i	0.357432	+	0.357432i
$870$		0			0
$871$	13.3315			0.0153060
$872$		0			0
$873$		0			0
$874$		0			0
$875$	879.929	−	879.929i	1.00563	−	1.00563i
$876$		0			0
$877$	664.587	−	664.587i	0.757796	−	0.757796i	−0.218125	−	0.975921i	$-0.569994\pi$
$877$	664.587	−	664.587i	0.757796	−	0.757796i	0.975921	+	0.218125i	$0.0699939\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−772.898			−0.877296			−0.438648	−	0.898659i	$-0.644542\pi$
$881$	−772.898			−0.877296			−0.438648	+	0.898659i	$0.644542\pi$
$882$		0			0
$883$	−321.603	−	321.603i	−0.364216	−	0.364216i	0.501146	−	0.865363i	$-0.332912\pi$
$883$	−321.603	−	321.603i	−0.364216	−	0.364216i	−0.865363	+	0.501146i	$0.832912\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	762.665			0.859825			0.429912	−	0.902871i	$-0.358544\pi$
$887$	762.665			0.859825			0.429912	+	0.902871i	$0.358544\pi$
$888$		0			0
$889$		−	511.245i		−	0.575079i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−39.2093	+	39.2093i	−0.0439074	+	0.0439074i
$894$		0			0
$895$		−	566.637i		−	0.633114i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	815.741	+	815.741i	0.907388	+	0.907388i
$900$		0			0
$901$	−459.533	−	459.533i	−0.510025	−	0.510025i
$902$		0			0
$903$		0			0
$904$		0			0
$905$			581.006i			0.641996i
$906$		0			0
$907$	−729.007	+	729.007i	−0.803756	+	0.803756i	−0.983680	−	0.179924i	$-0.942415\pi$
$907$	−729.007	+	729.007i	−0.803756	+	0.803756i	0.179924	+	0.983680i	$0.442415\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	885.541i		−	0.972054i	−0.873944	−	0.486027i	$-0.838446\pi$
$911$		−	885.541i		−	0.972054i	0.873944	−	0.486027i	$-0.161554\pi$
$912$		0			0
$913$	362.546			0.397094
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−467.230	−	467.230i	−0.509520	−	0.509520i
$918$		0			0
$919$	−714.964			−0.777980			−0.388990	−	0.921242i	$-0.627176\pi$
$919$	−714.964			−0.777980			−0.388990	+	0.921242i	$0.627176\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−97.9727	+	97.9727i	−0.106146	+	0.106146i
$924$		0			0
$925$	245.967	−	245.967i	0.265911	−	0.265911i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−585.585			−0.630339			−0.315170	−	0.949035i	$-0.602061\pi$
$929$	−585.585			−0.630339			−0.315170	+	0.949035i	$0.602061\pi$
$930$		0			0
$931$	−165.536	−	165.536i	−0.177805	−	0.177805i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−814.960			−0.871614
$936$		0			0
$937$			1423.91i			1.51965i	0.650129	+	0.759824i	$0.274715\pi$
$937$			1423.91i			1.51965i	−0.650129	+	0.759824i	$0.725285\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	142.471	−	142.471i	0.151404	−	0.151404i	−0.627341	−	0.778745i	$-0.715857\pi$
$941$	142.471	−	142.471i	0.151404	−	0.151404i	0.778745	+	0.627341i	$0.215857\pi$
$942$		0			0
$943$		−	2416.56i		−	2.56264i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−174.352	−	174.352i	−0.184110	−	0.184110i	0.609034	−	0.793144i	$-0.291557\pi$
$947$	−174.352	−	174.352i	−0.184110	−	0.184110i	−0.793144	+	0.609034i	$0.791557\pi$
$948$		0			0
$949$	43.7733	+	43.7733i	0.0461257	+	0.0461257i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	66.9080i		−	0.0702078i	−0.999384	−	0.0351039i	$-0.988824\pi$
$953$		−	66.9080i		−	0.0702078i	0.999384	−	0.0351039i	$-0.0111762\pi$
$954$		0			0
$955$	1115.53	−	1115.53i	1.16809	−	1.16809i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		−	620.870i		−	0.647414i
$960$		0			0
$961$	−95.4048			−0.0992766
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−1398.86	−	1398.86i	−1.44959	−	1.44959i
$966$		0			0
$967$	128.892			0.133290			0.0666451	−	0.997777i	$-0.478770\pi$
$967$	128.892			0.133290			0.0666451	+	0.997777i	$0.478770\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	801.089	−	801.089i	0.825015	−	0.825015i	−0.161808	−	0.986822i	$-0.551732\pi$
$971$	801.089	−	801.089i	0.825015	−	0.825015i	0.986822	+	0.161808i	$0.0517324\pi$
$972$		0			0
$973$	468.959	−	468.959i	0.481972	−	0.481972i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	1591.49			1.62896			0.814479	−	0.580194i	$-0.197023\pi$
$977$	1591.49			1.62896			0.814479	+	0.580194i	$0.197023\pi$
$978$		0			0
$979$	124.953	+	124.953i	0.127633	+	0.127633i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−1699.86			−1.72926			−0.864631	−	0.502408i	$-0.832448\pi$
$983$	−1699.86			−1.72926			−0.864631	+	0.502408i	$0.832448\pi$
$984$		0			0
$985$		−	238.167i		−	0.241794i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	703.939	−	703.939i	0.711769	−	0.711769i
$990$		0			0
$991$		−	228.875i		−	0.230954i	−0.993310	−	0.115477i	$-0.963160\pi$
$991$		−	228.875i		−	0.230954i	0.993310	−	0.115477i	$-0.0368397\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	1475.03	+	1475.03i	1.48244	+	1.48244i
$996$		0			0
$997$	−1020.63	−	1020.63i	−1.02370	−	1.02370i	−0.999712	−	0.0239926i	$-0.992362\pi$
$997$	−1020.63	−	1020.63i	−1.02370	−	1.02370i	−0.0239926	−	0.999712i	$-0.507638\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1152.3.m.c.415.8		16
3.2	odd	2		384.3.l.b.31.1		16
4.3	odd	2		1152.3.m.f.415.8		16
8.3	odd	2		144.3.m.c.91.7		16
8.5	even	2		576.3.m.c.271.1		16
12.11	even	2		384.3.l.a.31.5		16
16.3	odd	4	inner	1152.3.m.c.991.8		16
16.5	even	4		144.3.m.c.19.7		16
16.11	odd	4		576.3.m.c.559.1		16
16.13	even	4		1152.3.m.f.991.8		16
24.5	odd	2		192.3.l.a.79.8		16
24.11	even	2		48.3.l.a.43.2	yes	16
48.5	odd	4		48.3.l.a.19.2	✓	16
48.11	even	4		192.3.l.a.175.8		16
48.29	odd	4		384.3.l.a.223.5		16
48.35	even	4		384.3.l.b.223.1		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
48.3.l.a.19.2	✓	16	48.5	odd	4
48.3.l.a.43.2	yes	16	24.11	even	2
144.3.m.c.19.7		16	16.5	even	4
144.3.m.c.91.7		16	8.3	odd	2
192.3.l.a.79.8		16	24.5	odd	2
192.3.l.a.175.8		16	48.11	even	4
384.3.l.a.31.5		16	12.11	even	2
384.3.l.a.223.5		16	48.29	odd	4
384.3.l.b.31.1		16	3.2	odd	2
384.3.l.b.223.1		16	48.35	even	4
576.3.m.c.271.1		16	8.5	even	2
576.3.m.c.559.1		16	16.11	odd	4
1152.3.m.c.415.8		16	1.1	even	1	trivial
1152.3.m.c.991.8		16	16.3	odd	4	inner
1152.3.m.f.415.8		16	4.3	odd	2
1152.3.m.f.991.8		16	16.13	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1152 = 2^{7} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1152.m (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(6.49473 + 6.49473i) q^{5} -3.94273 q^{7} +O(q^{10})\)	\(q+(6.49473 + 6.49473i) q^{5} -3.94273 q^{7} +(-4.31091 + 4.31091i) q^{11} +(-4.06281 + 4.06281i) q^{13} +14.5538 q^{17} +(4.94805 + 4.94805i) q^{19} -43.6717 q^{23} +59.3629i q^{25} +(25.0979 - 25.0979i) q^{29} +32.5024i q^{31} +(-25.6069 - 25.6069i) q^{35} +(-4.14345 - 4.14345i) q^{37} +55.3348i q^{41} +(-16.1189 + 16.1189i) q^{43} +7.92420i q^{47} -33.4549 q^{49} +(-31.5748 - 31.5748i) q^{53} -55.9964 q^{55} +(49.7172 - 49.7172i) q^{59} +(-44.4711 + 44.4711i) q^{61} -52.7736 q^{65} +(-1.64068 - 1.64068i) q^{67} +24.1145 q^{71} -10.7741i q^{73} +(16.9967 - 16.9967i) q^{77} -72.0517i q^{79} +(-42.0499 - 42.0499i) q^{83} +(94.5229 + 94.5229i) q^{85} -28.9853i q^{89} +(16.0185 - 16.0185i) q^{91} +64.2724i q^{95} -54.2698 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q+O(q^{10}) \) 16 * q	\( 16 q - 32 q^{11} - 32 q^{19} - 128 q^{23} + 32 q^{29} - 96 q^{35} + 96 q^{37} + 160 q^{43} + 112 q^{49} - 160 q^{53} + 256 q^{55} + 128 q^{59} + 32 q^{61} + 32 q^{65} + 320 q^{67} + 512 q^{71} + 224 q^{77} + 160 q^{83} - 160 q^{85} - 480 q^{91}+O(q^{100}) \) 16 * q - 32 * q^11 - 32 * q^19 - 128 * q^23 + 32 * q^29 - 96 * q^35 + 96 * q^37 + 160 * q^43 + 112 * q^49 - 160 * q^53 + 256 * q^55 + 128 * q^59 + 32 * q^61 + 32 * q^65 + 320 * q^67 + 512 * q^71 + 224 * q^77 + 160 * q^83 - 160 * q^85 - 480 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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